Abstract In order to make a detailed analysis of the laser-induced breakdown spectroscopy (LIBS), we present a new approach which has advantages of singularity free and derivative free for solving the Abel integral equation. After presenting the model describing the LIBS, a collocation method is applied and two main steps are introduced. First, the Abel integral equation is transformed to a singularity free and derivative free integral equation. Then, by expanding the solution of the integral equation to a polynomial of degree n, the solution can be approximated by a linear system established by the collocation method. Our approach is suitable for dealing with LIBS problem for any form of the measured intensity. In particular, the method is insensitive to the noisy intensity data. A convergence analysis is also given to show the validity of the method and numerical examples are provided to show its efficiency and accuracy.

中文翻译：

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激光诱导击穿光谱分析中阿贝尔积分方程的无奇异点和无导数方法

摘要 为了对激光诱导击穿光谱（LIBS）进行详细的分析，提出了一种具有无奇异点和无导数优点的求解阿贝尔积分方程的新方法。在介绍了描述 LIBS 的模型之后，应用了搭配方法并介绍了两个主要步骤。首先，将阿贝尔积分方程转化为无奇异点和无导数的积分方程。然后，通过将积分方程的解展开为n次多项式，就可以用搭配法建立的线性系统来近似解。我们的方法适用于处理任何形式的测量强度的 LIBS 问题。特别是，该方法对噪声强度数据不敏感。